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#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\pdf_title "Accelerating lattice mode calculations with T-matrix method"
\pdf_author "Marek Nečada"
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\begin_body
\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\uoft}[1]{\mathfrak{F}#1}
\end_inset
\begin_inset FormulaMacro
\newcommand{\uaft}[1]{\mathfrak{\mathbb{F}}#1}
\end_inset
\begin_inset FormulaMacro
\newcommand{\usht}[2]{\mathbb{S}_{#1}#2}
\end_inset
\begin_inset FormulaMacro
\newcommand{\bsht}[2]{\mathrm{S}_{#1}#2}
\end_inset
\begin_inset FormulaMacro
\newcommand{\pht}[2]{\mathfrak{\mathbb{H}}_{#1}#2}
\end_inset
\begin_inset FormulaMacro
\newcommand{\vect}[1]{\mathbf{#1}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\ud}{\mathrm{d}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\basis}[1]{\mathfrak{#1}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\dc}[1]{Ш_{#1}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\rec}[1]{#1^{-1}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\recb}[1]{#1^{\widehat{-1}}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\ints}{\mathbb{Z}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\nats}{\mathbb{N}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\reals}{\mathbb{R}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\ush}[2]{Y_{#1,#2}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\hgfr}{\mathbf{F}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\hgf}{F}
\end_inset
\begin_inset FormulaMacro
\newcommand{\ph}{\mathrm{ph}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\kor}[1]{\underline{#1}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\koru}[1]{\utilde{#1}}
\end_inset
\end_layout
\begin_layout Title
Accelerating lattice mode calculations with
\begin_inset Formula $T$
\end_inset
-matrix method
\end_layout
\begin_layout Author
Marek Nečada
\end_layout
\begin_layout Abstract
The
\begin_inset Formula $T$
\end_inset
-matrix approach is the method of choice for simulating optical response
of a reasonably small system of compact linear scatterers on isotropic
background.
However, its direct utilisation for problems with infinite lattices is
problematic due to slowly converging sums over the lattice.
Here I develop a way to compute the problematic sums in the reciprocal
space, making the
\begin_inset Formula $T$
\end_inset
-matrix method very suitable for infinite periodic systems as well.
\end_layout
\begin_layout Section
Formulation of the problem
\end_layout
\begin_layout Standard
Assume a system of compact EM scatterers in otherwise homogeneous and isotropic
medium, and assume that the system, i.e.
both the medium and the scatterers, have linear response.
A scattering problem in such system can be written as
\begin_inset Formula
\[
A_{α}=T_{α}P_{α}=T_{α}(\sum_{β}S_{α\leftarrowβ}A_{β}+P_{0α})
\]
\end_inset
where
\begin_inset Formula $T_{α}$
\end_inset
is the
\begin_inset Formula $T$
\end_inset
-matrix for scatterer α,
\begin_inset Formula $A_{α}$
\end_inset
is its vector of the scattered wave expansion coefficient (the multipole
indices are not explicitely indicated here) and
\begin_inset Formula $P_{α}$
\end_inset
is the local expansion of the incoming sources.
\begin_inset Formula $S_{α\leftarrowβ}$
\end_inset
is ...
and ...
is ...
\end_layout
\begin_layout Standard
...
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\sum_{β}(\delta_{αβ}-T_{α}S_{α\leftarrowβ})A_{β}=T_{α}P_{0α}.
\]
\end_inset
\end_layout
\begin_layout Standard
Now suppose that the scatterers constitute an infinite lattice
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{\vect aα}S_{\vect aα\leftarrow\vect bβ})A_{\vect bβ}=T_{\vect aα}P_{0\vect aα}.
\]
\end_inset
Due to the periodicity, we can write
\begin_inset Formula $S_{\vect aα\leftarrow\vect bβ}=S_{α\leftarrowβ}(\vect b-\vect a)$
\end_inset
and
\begin_inset Formula $T_{\vect aα}=T_{\alpha}$
\end_inset
.
In order to find lattice modes, we search for solutions with zero RHS
\begin_inset Formula
\[
\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{α}S_{\vect aα\leftarrow\vect bβ})A_{\vect bβ}=0
\]
\end_inset
and we assume periodic solution
\begin_inset Formula $A_{\vect b\beta}(\vect k)=A_{\vect a\beta}e^{i\vect k\cdot\vect r_{\vect b-\vect a}}$
\end_inset
, yielding
\begin_inset Formula
\begin{eqnarray*}
\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{α}S_{\vect aα\leftarrow\vect bβ})A_{\vect a\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b-\vect a}} & = & 0,\\
\sum_{\vect bβ}(\delta_{\vect{0b}}\delta_{αβ}-T_{α}S_{\vect 0α\leftarrow\vect bβ})A_{\vect 0\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b}} & = & 0,\\
\sum_{β}(\delta_{αβ}-T_{α}\underbrace{\sum_{\vect b}S_{\vect 0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}}_{W_{\alpha\beta}(\vect k)})A_{\vect 0\beta}\left(\vect k\right) & = & 0,\\
A_{\vect 0\alpha}\left(\vect k\right)-T_{α}\sum_{\beta}W_{\alpha\beta}\left(\vect k\right)A_{\vect 0\beta}\left(\vect k\right) & = & 0.
\end{eqnarray*}
\end_inset
Therefore, in order to solve the modes, we need to compute the
\begin_inset Quotes eld
\end_inset
lattice Fourier transform
\begin_inset Quotes erd
\end_inset
of the translation operator,
\begin_inset Formula
\begin{equation}
W_{\alpha\beta}(\vect k)\equiv\sum_{\vect b}S_{\vect 0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}.\label{eq:W definition}
\end{equation}
\end_inset
\end_layout
\begin_layout Section
Computing the Fourier sum of the translation operator
\end_layout
\begin_layout Standard
The problem evaluating
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W definition"
\end_inset
is the asymptotic behaviour of the translation operator,
\begin_inset Formula $S_{\vect 0α\leftarrow\vect bβ}\sim\left|\vect r_{\vect b}\right|^{-1}e^{ik_{0}\left|\vect r_{\vect b}\right|}$
\end_inset
that makes the convergence of the sum quite problematic for any
\begin_inset Formula $d>1$
\end_inset
-dimensional lattice.
\begin_inset Foot
status open
\begin_layout Plain Layout
Note that
\begin_inset Formula $d$
\end_inset
here is dimensionality of the lattice, not the space it lies in, which
I for certain reasons assume to be three.
(TODO few notes on integration and reciprocal lattices in some appendix)
\end_layout
\end_inset
In electrostatics, one can solve this problem with Ewald summation.
Its basic idea is that if what asymptoticaly decays poorly in the direct
space, will perhaps decay fast in the Fourier space.
I use the same idea here, but everything will be somehow harder than in
electrostatics.
\end_layout
\begin_layout Standard
Let us re-express the sum in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W definition"
\end_inset
in terms of integral with a delta comb
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
W_{\alpha\beta}(\vect k)=\int\ud^{d}\vect r\dc{\basis u}(\vect r)S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})e^{i\vect k\cdot\vect r}.\label{eq:W integral}
\end{equation}
\end_inset
The translation operator
\begin_inset Formula $S$
\end_inset
is now a function defined in the whole 3d space;
\begin_inset Formula $\vect r_{\alpha},\vect r_{\beta}$
\end_inset
are the displacements of scatterers
\begin_inset Formula $\alpha$
\end_inset
and
\begin_inset Formula $\beta$
\end_inset
in a unit cell.
The arrow notation
\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})$
\end_inset
means
\begin_inset Quotes eld
\end_inset
translation operator for spherical waves originating in
\begin_inset Formula $\vect r+\vect r_{\beta}$
\end_inset
evaluated in
\begin_inset Formula $\vect r_{\alpha}$
\end_inset
\begin_inset Quotes erd
\end_inset
and obviously
\begin_inset Formula $S$
\end_inset
is in fact a function of a single 3d argument,
\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect 0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$
\end_inset
.
Expression
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W integral"
\end_inset
can be rewritten as
\begin_inset Formula
\[
W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0))\left(\vect k\right)}
\]
\end_inset
where changed the sign of
\begin_inset Formula $\vect r/\vect{\bullet}$
\end_inset
has been swapped under integration, utilising evenness of
\begin_inset Formula $\dc{\basis u}$
\end_inset
.
Fourier transform of product is convolution of Fourier transforms, so (using
formula
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Dirac comb uaFt"
\end_inset
for the Fourier transform of Dirac comb)
\begin_inset Formula
\begin{eqnarray}
W_{\alpha\beta}(\vect k) & = & \left(\left(\uaft{\dc{\basis u}}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)(\vect k)\nonumber \\
& = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\left(\dc{\recb{\basis u}}^{(d)}\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)\left(\vect k\right)\nonumber \\
& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right).\label{eq:W sum in reciprocal space}
\end{eqnarray}
\end_inset
\begin_inset Note Note
status open
\begin_layout Plain Layout
Factor
\begin_inset Formula $\left(2\pi\right)^{\frac{d}{2}}$
\end_inset
cancels out with the
\begin_inset Formula $\left(2\pi\right)^{-\frac{d}{2}}$
\end_inset
factor appearing in the convolution/product formula in the unitary angular
momentum convention.
\end_layout
\end_inset
As such, this is not extremely helpful because the the
\emph on
whole
\emph default
translation operator
\begin_inset Formula $S$
\end_inset
has singularities in origin, hence its Fourier transform
\begin_inset Formula $\uaft S$
\end_inset
will decay poorly.
\end_layout
\begin_layout Standard
However, Fourier transform is linear, so we can in principle separate
\begin_inset Formula $S$
\end_inset
in two parts,
\begin_inset Formula $S=S^{\textup{L}}+S^{\textup{S}}$
\end_inset
.
\begin_inset Formula $S^{\textup{S}}$
\end_inset
is a short-range part that decays sufficiently fast with distance so that
its direct-space lattice sum converges well;
\begin_inset Formula $S^{\textup{S}}$
\end_inset
must as well contain all the singularities of
\begin_inset Formula $S$
\end_inset
in the origin.
The other part,
\begin_inset Formula $S^{\textup{L}}$
\end_inset
, will retain all the slowly decaying terms of
\begin_inset Formula $S$
\end_inset
but it also has to be smooth enough in the origin, so that its Fourier
transform
\begin_inset Formula $\uaft{S^{\textup{L}}}$
\end_inset
decays fast enough.
(The same idea lies behind the Ewald summation in electrostatics.) Using
the linearity of Fourier transform and formulae
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W definition"
\end_inset
and
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W sum in reciprocal space"
\end_inset
, the operator
\begin_inset Formula $W_{\alpha\beta}$
\end_inset
can then be re-expressed as
\begin_inset Formula
\begin{eqnarray}
W_{\alpha\beta}\left(\vect k\right) & = & W_{\alpha\beta}^{\textup{S}}\left(\vect k\right)+W_{\alpha\beta}^{\textup{L}}\left(\vect k\right)\nonumber \\
W_{\alpha\beta}^{\textup{S}}\left(\vect k\right) & = & \sum_{\vect R\in\basis u\ints^{d}}S^{\textup{S}}(\vect 0\leftarrow\vect R+\vect r_{\beta}-\vect r_{\alpha})e^{i\vect k\cdot\vect R}\label{eq:W Short definition}\\
W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S^{\textup{L}}(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W Long definition}
\end{eqnarray}
\end_inset
where both sums should converge nicely.
\end_layout
\begin_layout Section
Finding a good decomposition
\end_layout
\begin_layout Standard
The remaining challenge is therefore finding a suitable decomposition
\begin_inset Formula $S^{\textup{L}}+S^{\textup{S}}$
\end_inset
such that both
\begin_inset Formula $S^{\textup{S}}$
\end_inset
and
\begin_inset Formula $\uaft{S^{\textup{L}}}$
\end_inset
decay fast enough with distance and are expressable analytically.
With these requirements, I do not expect to find gaussian asymptotics as
in the electrostatic Ewald formula—having
\begin_inset Formula $\sim x^{-t}$
\end_inset
,
\begin_inset Formula $t>d$
\end_inset
asymptotics would be nice, making the sums in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W Short definition"
\end_inset
,
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W Long definition"
\end_inset
absolutely convergent.
\end_layout
\begin_layout Standard
The translation operator
\begin_inset Formula $S$
\end_inset
for compact scatterers in 3d can be expressed as
\begin_inset Formula
\[
S_{l',m',t'\leftarrow l,m,t}\left(\vect r\leftarrow\vect 0\right)=\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect r},\phi_{\vect r}\right)z_{p}^{(J)}\left(k_{0}\left|\vect r\right|\right)
\]
\end_inset
where
\begin_inset Formula $Y_{l,m}\left(\theta,\phi\right)$
\end_inset
are the spherical harmonics,
\begin_inset Formula $z_{p}^{(J)}\left(r\right)$
\end_inset
some of the Bessel or Hankel functions (probably
\begin_inset Formula $h_{p}^{(1)}$
\end_inset
in all meaningful cases; TODO) and
\begin_inset Formula $c_{p}^{l,m,t\leftarrow l',m',t'}$
\end_inset
are some ugly but known coefficients (REF Xu 1996, eqs.
76,77).
\end_layout
\begin_layout Standard
The spherical Hankel functions can be expressed analytically as (REF DLMF
10.49.6, 10.49.1)
\begin_inset Formula
\begin{equation}
h_{n}^{(1)}(r)=e^{ir}\sum_{k=0}^{n}\frac{i^{k-n-1}}{r^{k+1}}\frac{\left(n+k\right)!}{2^{k}k!\left(n-k\right)!},\label{eq:spherical Hankel function series}
\end{equation}
\end_inset
so if we find a way to deal with the radial functions
\begin_inset Formula $s_{k_{0},q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
\end_inset
,
\begin_inset Formula $q=1,2$
\end_inset
in 2d case or
\begin_inset Formula $q=1,2,3$
\end_inset
in 3d case, we get absolutely convergent summations in the direct space.
\end_layout
\begin_layout Subsection
2d
\end_layout
\begin_layout Standard
Assume that all scatterers are placed in the plane
\begin_inset Formula $\vect z=0$
\end_inset
, so that the 2d Fourier transform of the long-range part of the translation
operator in terms of Hankel transforms, according to
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Fourier v. Hankel tf 2d"
\end_inset
, reads
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{multline*}
\uaft{S_{l',m',t'\leftarrow l,m,t}^{\textup{L}}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\\
\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\frac{\pi}{2},0\right)e^{i(m'-m)\phi}i^{m'-m}\pht{m'-m}{h_{p}^{(1)\textup{L}}\left(k_{0}\vect{\bullet}\right)}\left(\left|\vect k\right|\right)
\end{multline*}
\end_inset
Here
\begin_inset Formula $h_{p}^{(1)\textup{L}}=h_{p}^{(1)}-h_{p}^{(1)\textup{S}}$
\end_inset
is a long range part of a given spherical Hankel function which has to
be found and which contains all the terms with far-field (
\begin_inset Formula $r\to\infty$
\end_inset
) asymptotics proportional to
\begin_inset Formula $\sim e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
\end_inset
,
\begin_inset Formula $q\le Q$
\end_inset
where
\begin_inset Formula $Q$
\end_inset
is at least two in order to achieve absolute convergence of the direct-space
sum, but might be higher in order to speed the convergence up.
\end_layout
\begin_layout Standard
Obviously, all the terms
\begin_inset Formula $\propto s_{k_{0},q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
\end_inset
,
\begin_inset Formula $q>Q$
\end_inset
of the spherical Hankel function
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:spherical Hankel function series"
\end_inset
can be kept untouched as part of
\begin_inset Formula $h_{p}^{(1)\textup{S}}$
\end_inset
, as they decay fast enough.
\end_layout
\begin_layout Standard
The remaining task is therefore to find a suitable decomposition of
\begin_inset Formula $s_{k_{0},q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
\end_inset
,
\begin_inset Formula $q\le Q$
\end_inset
into short-range and long-range parts,
\begin_inset Formula $s_{k_{0},q}(r)=s_{k_{0},q}^{\textup{S}}(r)+s_{k_{0},q}^{\textup{L}}(r)$
\end_inset
, such that
\begin_inset Formula $s_{k_{0},q}^{\textup{L}}(r)$
\end_inset
contains all the slowly decaying asymptotics and its Hankel transforms
decay desirably fast as well,
\begin_inset Formula $\pht n{s_{k_{0},q}^{\textup{L}}}\left(k\right)=o(z^{-Q})$
\end_inset
,
\begin_inset Formula $z\to\infty$
\end_inset
.
The latter requirement calls for suitable regularisation functions—
\begin_inset Formula $s_{q}^{\textup{L}}$
\end_inset
must be sufficiently smooth in the origin, so that
\begin_inset Formula
\begin{equation}
\pht n{s_{k_{0},q}^{\textup{L}}}\left(k\right)=\int_{0}^{\infty}s_{k_{0},q}^{\textup{L}}\left(r\right)rJ_{n}\left(kr\right)\ud r=\int_{0}^{\infty}s_{k_{0},q}\left(r\right)\rho\left(r\right)rJ_{n}\left(kr\right)\ud r\label{eq:2d long range regularisation problem statement}
\end{equation}
\end_inset
exists and decays fast enough.
\begin_inset Formula $J_{\nu}(r)\sim\left(r/2\right)^{\nu}/\Gamma\left(\nu+1\right)$
\end_inset
(REF DLMF 10.7.3) near the origin, so the regularisation function should
be
\begin_inset Formula $\rho(r)=o(r^{q-n-1})$
\end_inset
only to make
\begin_inset Formula $\pht n{s_{q}^{\textup{L}}}$
\end_inset
converge.
The additional decay speed requirement calls for at least
\begin_inset Formula $\rho(r)=o(r^{q-n+Q-1})$
\end_inset
, I guess.
At the same time,
\begin_inset Formula $\rho(r)$
\end_inset
must converge fast enough to one for
\begin_inset Formula $r\to\infty$
\end_inset
.
\end_layout
\begin_layout Standard
The electrostatic Ewald summation uses regularisation with
\begin_inset Formula $1-e^{-cr^{2}}$
\end_inset
.
However, such choice does not seem to lead to an analytical solution (really?
could not something be dug out of DLMF 10.22.54?) for the current problem
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:2d long range regularisation problem statement"
\end_inset
.
But it turns out that the family of functions
\begin_inset Formula
\begin{equation}
\rho_{\kappa,c}(r)\equiv\left(1-e^{-cr}\right)^{\text{\kappa}},\quad c>0,\kappa\in\nats\label{eq:binom regularisation function}
\end{equation}
\end_inset
might lead to satisfactory results; see below.
\begin_inset Note Note
status open
\begin_layout Plain Layout
Another analytically feasible possibility could be
\begin_inset Formula
\begin{equation}
\rho_{p}^{\textup{ig.}}\equiv e^{-p/x^{2}}.\label{eq:inverse gaussian regularisation function}
\end{equation}
\end_inset
\end_layout
\begin_layout Plain Layout
Nope, propably did not work.
\end_layout
\end_inset
\end_layout
\begin_layout Subsubsection
Hankel transforms of the long-range parts, „binomial“ regularisation
\begin_inset CommandInset label
LatexCommand label
name "sub:Hankel-transforms-binom-reg"
\end_inset
\end_layout
\begin_layout Standard
Let
\begin_inset Note Note
status open
\begin_layout Plain Layout
\begin_inset Formula $\rho_{\kappa,c}$
\end_inset
from
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:binom regularisation function"
\end_inset
serve as the regularisation fuction and
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{eqnarray}
\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & \equiv & \int_{0}^{\infty}\frac{e^{ik_{0}r}}{\left(k_{0}r\right)^{q}}J_{n}\left(kr\right)\left(1-e^{-cr}\right)^{\kappa}r\,\ud r\nonumber \\
& = & k_{0}^{-q}\int_{0}^{\infty}r^{1-q}J_{n}\left(kr\right)\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}e^{-(\sigma c-ik_{0})r}\ud r\nonumber \\
& \underset{\equiv}{\textup{form.}} & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\pht n{s_{q,k_{0}}^{\textup{L}1,\sigma c}}\left(k\right).\label{eq:2D Hankel transform of regularized outgoing wave, decomposition}
\end{eqnarray}
\end_inset
From [REF DLMF 10.22.49] one digs
\begin_inset Note Note
status open
\begin_layout Plain Layout
\begin_inset Formula
\begin{eqnarray*}
\mu & \leftarrow & 2-q\\
\nu & \leftarrow & n\\
b & \leftarrow & k\\
a & \leftarrow & c-ik_{0}
\end{eqnarray*}
\end_inset
\end_layout
\end_inset
\begin_inset Formula
\begin{multline}
\pht n{s_{q,k_{0}}^{\textup{L}1,c}}\left(k\right)=\frac{k^{n}Γ\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(c-ik_{0}\right)^{2-q+n}}\hgfr\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right),\\
\Re\left(2-q+n\right)>0,\Re(c-ik_{0}\pm k)\ge0,\label{eq:2D Hankel transform of exponentially suppressed outgoing wave as 2F1}
\end{multline}
\end_inset
and using [REF DLMF 15.9.17] and
\begin_inset Note Note
status open
\begin_layout Plain Layout
\begin_inset Formula $P_{\nu}^{\mu}=P_{-\nu-1}^{\mu}$
\end_inset
\end_layout
\end_inset
[REF DLMF 14.9.5]
\end_layout
\begin_layout Standard
\begin_inset Note Note
status collapsed
\begin_layout Plain Layout
\begin_inset Formula
\begin{eqnarray*}
\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\hgfr\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)\\
\mbox{(D15.2.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}Γ\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\sum_{s=0}^{\infty}\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{3-q+n}{2}\right)_{s}}{Γ(1+n+s)s!}\left(\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{s},\quad\left|\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right|<1\\
\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Formula
\begin{eqnarray*}
\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\hgfr\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)\\
\mbox{(D15.8.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}(\\
& & \pi\frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}{Γ\left(\frac{3-q+n}{2}\right)\text{Γ}\left(1+n-\frac{2-q+n}{2}\right)}\hgfr\left(\begin{array}{c}
\frac{2-q+n}{2},\frac{2-q+n}{2}-\left(1+n\right)+1\\
1/2
\end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)\\
& - & \pi\frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{3-q+n}{2}}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(1+n-\frac{3-q+n}{2}\right)}\hgfr\left(\begin{array}{c}
\frac{3-q+n}{2},\frac{3-q+n}{2}-\left(1+n\right)+1\\
3/2
\end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right))\\
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\pi(\\
& & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)}\hgfr\left(\begin{array}{c}
\frac{2-q+n}{2},\frac{2-q-n}{2}\\
1/2
\end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)\\
& - & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{3-q+n}{2}}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)}\hgfr\left(\begin{array}{c}
\frac{3-q+n}{2},\frac{3-q-n}{2}\\
3/2
\end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right))\\
\mbox{(D15.2.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\pi\sum_{s=0}^{\infty}(\\
& & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)}\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{1}{2}+s\right)s!}\left(-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)^{s}\\
& - & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{3-q+n}{2}}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)}\frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3}{2}+s\right)s!}\left(-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)^{s})\\
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\kor{k^{n}}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\kor{\left(\sigma c-ik_{0}\right)^{2-q+n}}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}(\\
& & \frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}k^{-2+q\kor{-n}-2s}\left(\sigma c-ik_{0}\right)^{\kor{2-q+n}+2s}\\
& - & \frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}k^{-3+q\kor{-n}-2s}\left(\sigma c-ik_{0}\right)^{\kor{3-q+n}+2s})\\
\mbox{} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}(\\
& & \frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}\kor{k^{-2+q-2s}}\kor{\left(\sigma c-ik_{0}\right)^{2s}}\\
& - & \frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\kor{k^{-3+q-2s}}\kor{\left(\sigma c-ik_{0}\right)^{1+2s}})\\
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\\
& & \times\left(\underbrace{\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}}_{\equiv c_{q,n,s}}-\underbrace{\frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}}_{č_{q,n,s}}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\kor{\left(\sigma c-ik_{0}\right)^{2s}}c_{q,n,s}-\frac{\left(\sigma c-ik_{0}\right)^{2s+1}}{k}č_{q,n,s}\right)\\
\mbox{(binom.)} & = & \kor{\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(c_{q,n,s}\sum_{t=0}^{2s}\binom{2s}{t}\left(\kor{\sigma}c\right)^{t}\left(-ik_{0}\right)^{2s-t}-č_{q,n,s}\sum_{t=0}^{2s+1}\binom{2s+1}{t}\left(\kor{\sigma}c\right)^{t}\left(-ik_{0}\right)^{2s+1-t}k^{-1}\right)\\
\mbox{(conds?)} & = & \frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\kappa!\left(-1\right)^{\kappa}\left(c_{q,n,s}\sum_{t=0}^{2s}\binom{2s}{t}\begin{Bmatrix}t\\
\kappa
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s-t}-č_{q,n,s}\sum_{t=0}^{2s+1}\binom{2s+1}{t}\begin{Bmatrix}t\\
\kappa
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s+1-t}k^{-1}\right)
\end{eqnarray*}
\end_inset
now the Stirling number of the 2nd kind
\begin_inset Formula $\begin{Bmatrix}t\\
\kappa
\end{Bmatrix}=0$
\end_inset
if
\begin_inset Formula $\kappa>t$
\end_inset
.
\end_layout
\begin_layout Plain Layout
What about the gamma fn on the left? Using DLMF 5.5.5, which says
\begin_inset Formula $Γ(2z)=\pi^{-1/2}2^{2z-1}\text{Γ}(z)\text{Γ}(z+\frac{1}{2})$
\end_inset
we have
\begin_inset Formula
\[
\text{Γ}\left(2-q+n\right)=\frac{2^{1-q+n}}{\sqrt{\pi}}\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{3-q+n}{2}\right),
\]
\end_inset
so
\begin_inset Formula
\begin{eqnarray*}
\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \frac{\kor{\text{Γ}\left(2-q+n\right)}}{\kor{2^{n}}k_{0}^{q}}\kor{\pi}\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\kappa!\left(-1\right)^{\kappa}\left(\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\kor{\text{Γ}\left(\frac{3-q+n}{2}\right)}\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}\sum_{t=0}^{2s}\binom{2s}{t}\begin{Bmatrix}t\\
\kappa
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s-t}-\frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\kor{\text{Γ}\left(\frac{2-q+n}{2}\right)}\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\sum_{t=0}^{2s}\binom{2s+1}{t}\begin{Bmatrix}t\\
\kappa
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s+1-t}k^{-1}\right)\\
& = & \frac{2^{1-q}}{k_{0}^{q}}\sqrt{\pi}\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\kappa!\left(-1\right)^{\kappa}\left(\frac{\kor{\text{Γ}\left(\frac{2-q+n}{2}\right)\left(\frac{2-q+n}{2}\right)_{s}}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}\sum_{t=0}^{2s}\binom{2s}{t}\begin{Bmatrix}t\\
\kappa
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s-t}-\frac{\kor{\text{Γ}\left(\frac{3-q+n}{2}\right)\left(\frac{3-q+n}{2}\right)_{s}}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\sum_{t=0}^{2s}\binom{2s+1}{t}\begin{Bmatrix}t\\
\kappa
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s+1-t}k^{-1}\right)\\
\mbox{(D5.2.5)} & = & \frac{2^{1-q}}{k_{0}^{q}}\sqrt{\pi}\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\kappa!\left(-1\right)^{\kappa}\left(\frac{\text{Γ}\left(\frac{2-q+n}{2}+s\right)\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}\sum_{t=0}^{2s}\binom{2s}{t}\begin{Bmatrix}t\\
\kappa
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s-t}-\frac{\text{Γ}\left(\frac{3-q+n}{2}+s\right)\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\sum_{t=0}^{2s}\binom{2s+1}{t}\begin{Bmatrix}t\\
\kappa
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s+1-t}k^{-1}\right)
\end{eqnarray*}
\end_inset
The two terms have to be treated fifferently depending on whether q
\begin_inset Formula $q+n$
\end_inset
is even or odd.
\end_layout
\begin_layout Plain Layout
First, assume that
\begin_inset Formula $q+n$
\end_inset
is even, so the left term has gamma functions and pochhammer symbols with
integer arguments, while the right one has half-integer arguments.
As
\begin_inset Formula $n$
\end_inset
is non-negative and
\begin_inset Formula $q$
\end_inset
is positive,
\begin_inset Formula $\frac{q+n}{2}$
\end_inset
is positive, and the Pochhammer symbol
\begin_inset Formula $\left(\frac{2-q-n}{2}\right)_{s}=0$
\end_inset
if
\begin_inset Formula $s\ge\frac{q+n}{2}$
\end_inset
, which transforms the sum over
\begin_inset Formula $s$
\end_inset
to a finite sum for the left term.
However, there still remain divergent terms if
\begin_inset Formula $\frac{2-q+n}{2}+s\le0$
\end_inset
(let's handle this later; maybe D15.8.67 may be then be useful)! Now we
need to perform some transformations of variables to make the other sum
finite as well
\end_layout
\begin_layout Plain Layout
Pár kroků zpět:
\begin_inset Formula
\begin{eqnarray*}
\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\kor{\text{Γ}\left(2-q+n\right)}}{\kor{2^{n}}k_{0}^{q}}\kor{\pi}\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\times\left(\underbrace{\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\kor{\text{Γ}\left(\frac{3-q+n}{2}\right)}\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}}_{\equiv c_{q,n,s}}-\underbrace{\frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\kor{\text{Γ}\left(\frac{2-q+n}{2}\right)}\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}}_{č_{q,n,s}}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{1-q}}{k_{0}^{q}}\sqrt{\pi}\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\times\left(\frac{\text{Γ}\left(\frac{2-q+n}{2}\right)\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}-\frac{\text{Γ}\left(\frac{3-q+n}{2}\right)\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)
\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Plain Layout
If
\begin_inset Formula $q+n$
\end_inset
is even and
\begin_inset Formula $2-q+n\le0$
\end_inset
\begin_inset Formula
\begin{eqnarray*}
\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\kor{\hgfr}\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)\\
\mbox{(D15.1.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)\koru{\text{Γ}(1+n)}}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\koru{\hgf}\left(\frac{2-q+n}{2},\kor{\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}}\right)\\
\mbox{(D15.8.6)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\kor{k^{n}}\text{Γ}\left(2-q+n\right)\text{Γ}(1+n)}{2^{n}k_{0}^{q}\kor{\left(\sigma c-ik_{0}\right)^{2-q+n}}}\koru{\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\kor{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}}\hgf\left(\begin{array}{c}
\frac{2-q+n}{2},\koru{\kor{1-\left(1+n\right)+\frac{2-q+n}{2}}}\\
\koru{\kor{1-\frac{3-q+n}{2}+\frac{2-q+n}{2}}}
\end{array};\koru{\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}}\right)\\
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\koru{k^{q-2}}\text{Γ}\left(2-q+n\right)\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\koru{\frac{3}{2}\left(2-q+n\right)}}}\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\kor{\hgf\left(\begin{array}{c}
\frac{2-q+n}{2},\koru{\frac{2-q-n}{2}}\\
\koru{1/2}
\end{array};\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}\right)}\\
\mbox{(D15.2.1)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\kor{\text{Γ}\left(2-q+n\right)}\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\koru{\sum_{s=0}^{\infty}\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\left(\frac{1}{2}\right)_{s}s!}\left(\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}\right)^{s}}\\
\mbox{(D5.5.5)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{\kor{2^{n}}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\koru{\frac{2^{1-q\kor{+n}}}{\sqrt{\pi}}\kor{\text{Γ}\left(\frac{2-q+n}{2}\right)}\text{Γ}\left(\frac{3-q+n}{2}\right)}\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\sum_{s=0}^{\infty}\frac{\kor{\left(\frac{2-q+n}{2}\right)_{s}}\left(\frac{2-q-n}{2}\right)_{s}}{\left(\frac{1}{2}\right)_{s}s!}\left(\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}\right)^{s}\\
\mbox{(D5.2.5)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\koru{2^{1-q}}}{\sqrt{\pi}}\text{Γ}\left(\frac{3-q+n}{2}\right)\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\sum_{s=0}^{\infty}\frac{\koru{\text{Γ}\left(\frac{2-q+n}{2}+s\right)}\left(\frac{2-q-n}{2}\right)_{s}}{\left(\frac{1}{2}\right)_{s}s!}\left(\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}\right)^{s}\\
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{2^{1-q}}{\sqrt{\pi}}\text{Γ}\left(\frac{3-q+n}{2}\right)\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\sum_{s=0}^{\frac{q+n}{2}}\frac{\text{Γ}\left(\frac{2-q+n}{2}+s\right)\left(\frac{2-q-n}{2}\right)_{s}}{\left(\frac{1}{2}\right)_{s}s!}\left(\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}\right)^{s}\\
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{2^{1-q}}{\sqrt{\pi}}\text{Γ}\left(\frac{3-q+n}{2}\right)\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\sum_{s=0}^{\frac{q+n}{2}}\frac{\text{Γ}\left(\frac{2-q+n}{2}+s\right)\left(\frac{2-q-n}{2}\right)_{s}}{\left(\frac{1}{2}\right)_{s}s!}\left(\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}\right)^{s}
\end{eqnarray*}
\end_inset
now
\begin_inset Formula $\left(\frac{2-q-n}{2}\right)_{s}=0$
\end_inset
whenever
\begin_inset Formula $s\ge\frac{q+n}{2}$
\end_inset
and
\begin_inset Formula $\text{Γ}\left(\frac{2-q+n}{2}+s\right)$
\end_inset
is singular whenever
\begin_inset Formula $s\le-\frac{2-q+n}{2}$
\end_inset
, so we are no less fucked than before.
Maybe let's try the other variable transformation.
Or what about (D15.8.27)?
\begin_inset Formula
\begin{eqnarray*}
\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}\left(2-q+n\right)\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\kor{\hgf\left(\begin{array}{c}
\frac{2-q+n}{2},\frac{2-q-n}{2}\\
1/2
\end{array};\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}\right)}\\
\mbox{(D15.8.27)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}\left(2-q+n\right)\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\kor{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\koru{\frac{\kor{Γ\left(\frac{3-q+n}{2}\right)}Γ\left(\frac{3-q-n}{2}\right)}{2Γ\left(\frac{1}{2}\right)Γ\left(2-q+\frac{1}{2}\right)}\left(\hgf\left(\begin{array}{c}
2-q+n,2-q-n\\
2-q+\frac{1}{2}
\end{array};\frac{1}{2}-\frac{\sigma c-ik_{0}}{ik}\right)+\hgf\left(\begin{array}{c}
2-q+n,2-q-n\\
2-q+\frac{1}{2}
\end{array};\frac{1}{2}+\frac{\sigma c-ik_{0}}{ik}\right)\right)}\\
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}\left(2-q+n\right)\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\kor{\text{Γ}\koru{\left(\frac{3-q+n}{2}-\frac{2-q+n}{2}\right)}}\text{Γ}\left(\frac{3-q-n}{2}\right)}{\left(1+n\right)_{-\frac{2-q+n}{2}}2\kor{\text{Γ}\left(\frac{1}{2}\right)}\text{Γ}\left(2-q+\frac{1}{2}\right)}\left(\hgf\left(\begin{array}{c}
2-q+n,2-q-n\\
2-q+\frac{1}{2}
\end{array};\frac{1}{2}-\frac{\sigma c-ik_{0}}{ik}\right)+\hgf\left(\begin{array}{c}
2-q+n,2-q-n\\
2-q+\frac{1}{2}
\end{array};\frac{1}{2}+\frac{\sigma c-ik_{0}}{ik}\right)\right)\\
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}\left(2-q+n\right)\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\text{Γ}\left(\frac{3-q-n}{2}\right)}{\left(1+n\right)_{-\frac{2-q+n}{2}}2\text{Γ}\left(2-q+\frac{1}{2}\right)}\kor{\left(\hgf\left(\begin{array}{c}
2-q+n,2-q-n\\
2-q+\frac{1}{2}
\end{array};\frac{1}{2}-\frac{\sigma c-ik_{0}}{ik}\right)+\hgf\left(\begin{array}{c}
2-q+n,2-q-n\\
2-q+\frac{1}{2}
\end{array};\frac{1}{2}+\frac{\sigma c-ik_{0}}{ik}\right)\right)}\\
\mbox{(D15.2.1)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}\left(2-q+n\right)\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\text{Γ}\left(\frac{3-q-n}{2}\right)}{\left(1+n\right)_{-\frac{2-q+n}{2}}2\text{Γ}\left(2-q+\frac{1}{2}\right)}\koru{\sum_{s=0}^{\infty}\left(\frac{\left(2-q+n\right)_{s}\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\kor{\left(\left(\frac{1}{2}-\frac{\sigma c-ik_{0}}{ik}\right)^{s}+\left(\frac{1}{2}+\frac{\sigma c-ik_{0}}{ik}\right)^{s}\right)}\right)}\\
\mbox{(binom)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\text{Γ}\left(\frac{3-q-n}{2}\right)}{\left(1+n\right)_{-\frac{2-q+n}{2}}2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\kor{\left(2-q+n\right)_{s}}\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\koru{\sum_{r=0}^{s}\binom{s}{r}\left(\frac{\sigma c-ik_{0}}{ik}\right)^{r}2^{r-s}\left(\left(-1\right)^{r}+1\right)}\\
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\text{Γ}\left(\frac{3-q-n}{2}\right)}{\kor{\left(1+n\right)_{-\frac{2-q+n}{2}}}2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\koru{\text{Γ}\left(2-q+n+s\right)}\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\left(\frac{\sigma c-ik_{0}}{ik}\right)^{r}2^{r-s}\left(\left(-1\right)^{r}+1\right)\\
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}\kor{\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}}\frac{\koru{\text{Γ}\left(1+n\right)}\text{Γ}\left(\frac{3-q-n}{2}\right)}{\koru{\text{Γ}\left(\frac{q+n}{2}\right)}2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\kor{\left(\frac{\sigma c-ik_{0}}{ik}\right)^{r}}2^{r-s}\left(\left(-1\right)^{r}+1\right)\\
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}}\frac{\text{Γ}\left(1+n\right)\text{Γ}\left(\frac{3-q-n}{2}\right)}{Γ\left(\frac{q+n}{2}\right)2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\koru{\left(ik\right)^{-r}}\koru{\kor{\left(\sigma c-ik_{0}\right)^{r-\frac{3}{2}\left(2-q+n\right)}}}2^{r-s}\left(\left(-1\right)^{r}+1\right)\\
(bionm) & = & \kor{\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}}\frac{\text{Γ}\left(1+n\right)\text{Γ}\left(\frac{3-q-n}{2}\right)}{\text{Γ}\left(\frac{q+n}{2}\right)2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\left(ik\right)^{-r}\koru{\sum_{w=0}^{\infty|r-\frac{3}{2}\left(2-q+n\right)}\binom{r-\frac{3}{2}\left(2-q+n\right)}{w}\kor{\sigma^{w}}c^{w}\left(-ik_{0}^{r-\frac{3}{2}\left(2-q+n\right)-w}\right)}2^{r-s}\left(\left(-1\right)^{r}+1\right)\\
& = & \koru{\kappa!\left(-1\right)^{\kappa}}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}}\frac{\text{Γ}\left(1+n\right)\text{Γ}\left(\frac{3-q-n}{2}\right)}{\text{Γ}\left(\frac{q+n}{2}\right)2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=\kor 0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=\kor 0}^{s}\binom{\kor s}{\kor r}\left(ik\right)^{-r}\sum_{w=\kor 0}^{\infty|r-\frac{3}{2}\left(2-q+n\right)}\binom{r-\frac{3}{2}\left(2-q+n\right)}{\kor w}\koru{\kor{\begin{Bmatrix}w\\
\kappa
\end{Bmatrix}}}c^{w}\left(-ik_{0}^{r-\frac{3}{2}\left(2-q+n\right)-w}\right)2^{r-s}\left(\left(-1\right)^{r}+1\right)\\
& = & \kappa!\left(-1\right)^{\kappa}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}}\frac{\text{Γ}\left(1+n\right)\text{Γ}\left(\frac{3-q-n}{2}\right)}{\text{Γ}\left(\frac{q+n}{2}\right)2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=\koru{\kappa}}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=\koru{\kappa}}^{s}\binom{s}{r}\left(ik\right)^{-r}\sum_{w=\koru{\kappa}}^{\infty|r-\frac{3}{2}\left(2-q+n\right)}\binom{r-\frac{3}{2}\left(2-q+n\right)}{w}\begin{Bmatrix}w\\
\kappa
\end{Bmatrix}c^{w}\left(-ik_{0}^{r-\frac{3}{2}\left(2-q+n\right)-w}\right)2^{r-s}\left(\left(-1\right)^{r}+1\right)\\
& = & \kappa!\left(-1\right)^{\kappa}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}}\frac{\text{Γ}\left(1+n\right)\text{Γ}\left(\frac{3-q-n}{2}\right)}{\text{Γ}\left(\frac{q+n}{2}\right)2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=\kappa}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=\kappa}^{s}\binom{s}{r}\left(ik\right)^{-r}\sum_{w=\kappa}^{\infty|r-\frac{3}{2}\left(2-q+n\right)}\binom{r-\frac{3}{2}\left(2-q+n\right)}{w}\begin{Bmatrix}w\\
\kappa
\end{Bmatrix}c^{w}\left(-ik_{0}^{r-\frac{3}{2}\left(2-q+n\right)-w}\right)2^{r-s}\left(\left(-1\right)^{r}+1\right)
\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Plain Layout
The previous things are valid only if
\begin_inset Formula $q$
\end_inset
has a small non-integer part,
\begin_inset Formula $q=q'+\varepsilon$
\end_inset
.
They might still play a role in the series (especially in the infinite
ones) when taking the limit
\begin_inset Formula $\varepsilon\to0$
\end_inset
.
However, we got rid of the singularities in
\begin_inset Formula $\text{Γ}\left(2-q+n+s\right)$
\end_inset
if
\begin_inset Formula $\kappa$
\end_inset
is large enough.
\end_layout
\begin_layout Plain Layout
and we get same shit as before due to the singular
\begin_inset Formula $\text{Γ}\left(2-q+n+s\right)$
\end_inset
.
However,
\begin_inset Formula
\begin{eqnarray*}
(...) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\text{Γ}\left(\frac{3-q-n}{2}\right)}{\left(1+n\right)_{-\frac{2-q+n}{2}}2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\left(\frac{\sigma c-ik_{0}}{ik}\right)^{r}2^{r-s}\kor{\left(\left(-1\right)^{r}+1\right)}\\
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\text{Γ}\left(\frac{3-q-n}{2}\right)}{\left(1+n\right)_{-\frac{2-q+n}{2}}2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{\koru{floor(s/2)}}\binom{s}{\koru{2r}}\left(\frac{\sigma c-ik_{0}}{ik}\right)^{\koru{2r}}2^{\koru{2r}-s}\left(\left(-1\right)^{\koru{2r}}+1\right)
\end{eqnarray*}
\end_inset
\begin_inset Formula
\begin{eqnarray*}
(...) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\text{Γ}\left(\frac{3-q-n}{2}\right)}{\left(1+n\right)_{-\frac{2-q+n}{2}}2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\kor{\left(\frac{\sigma c-ik_{0}}{ik}\right)^{r}}2^{r-s}\left(\left(-1\right)^{r}+1\right)\\
binom & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\text{Γ}\left(\frac{3-q-n}{2}\right)}{\left(1+n\right)_{-\frac{2-q+n}{2}}2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\koru{\left(ik\right)^{-r}\sum_{b=0}^{r}\binom{r}{b}\sigma^{b}c^{b}\left(-ik_{0}\right)^{r-b}}2^{r-s}\left(\left(-1\right)^{r}+1\right)\\
& =
\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Plain Layout
aaah.
Let's assume that
\begin_inset Formula $q$
\end_inset
is not exactly
\begin_inset Formula
\begin{eqnarray*}
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\kor{\text{Γ}\left(2-q+n\right)}\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\sum_{s=0}^{\infty}\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\left(\frac{1}{2}\right)_{s}s!}\left(\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}\right)^{s}\\
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}\left(2-q+n\right)\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\sum_{s=0}^{\infty}k^{-2s}\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\left(\frac{1}{2}\right)_{s}s!}\left(\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}\right)^{s}
\end{eqnarray*}
\end_inset
zpět
\end_layout
\begin_layout Plain Layout
\begin_inset Formula
\begin{eqnarray*}
& = & \frac{2^{1-q}}{k_{0}^{q}}\sqrt{\pi}\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\kappa!\left(-1\right)^{\kappa}\left(\frac{\text{Γ}\left(\frac{2-q+n}{2}\right)\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}\sum_{t=0}^{2s}\binom{2s}{t}\begin{Bmatrix}t\\
\kappa
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s-t}-\frac{\text{Γ}\left(\frac{3-q+n}{2}\right)\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\sum_{t=0}^{2s+1}\binom{2s+1}{t}\begin{Bmatrix}t\\
\kappa
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s+1-t}k^{-1}\right)\\
& = & \frac{2^{1-q}}{k_{0}^{q}}\sqrt{\pi}\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\kappa!\left(-1\right)^{\kappa}\left(\frac{\text{Γ}\left(\frac{2-q+n}{2}\right)\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)\text{Γ}\left(1+s\right)}\sum_{t=0}^{2s}\binom{2s}{t}\begin{Bmatrix}t\\
\kappa
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s-t}-\frac{\text{Γ}\left(\frac{3-q+n}{2}\right)\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)\text{Γ}\left(1+s\right)}\sum_{t=0}^{2s+1}\binom{2s+1}{t}\begin{Bmatrix}t\\
\kappa
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s+1-t}k^{-1}\right)
\end{eqnarray*}
\end_inset
\end_layout
\end_inset
\begin_inset Note Note
status collapsed
\begin_layout Plain Layout
\begin_inset Formula
\begin{eqnarray*}
a & \leftarrow & \frac{2-q+n}{2}\\
c & \leftarrow & 1+n\\
z & \leftarrow & \frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}
\end{eqnarray*}
\end_inset
\begin_inset Formula
\begin{eqnarray*}
\pht n{s_{q,k_{0}}^{\textup{L}1,c}}\left(k\right) & = & \frac{k^{n}Γ\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(c-ik_{0}\right)^{2-q+n}}2^{n}\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{-\frac{n}{2}}\left(1-\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)\right)^{-\frac{2-q+n}{2}+\frac{n}{2}}P_{2-q+n-(1+n)}^{1-(1+n)}\left(\frac{1}{\sqrt{1-\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)}}\right)\\
& = & \frac{k^{n}Γ\left(2-q+n\right)}{k_{0}^{q}\left(c-ik_{0}\right)^{2-q+n}}\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{-\frac{n}{2}}\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{\frac{q}{2}-1}P_{1-q}^{-n}\left(\frac{1}{\sqrt{1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}}}\right)
\end{eqnarray*}
\end_inset
\begin_inset Formula
\[
\left|\ph\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right|<\pi,\quad\left|\ph\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)\right|<\pi
\]
\end_inset
in other words, neither
\begin_inset Formula $-k^{2}/\left(c-ik_{0}\right)^{2}$
\end_inset
nor
\begin_inset Formula $1+k^{2}/\left(c-ik_{0}\right)^{2}$
\end_inset
can be non-positive real number.
For assumed positive
\begin_inset Formula $k_{0}$
\end_inset
and non-negative
\begin_inset Formula $c$
\end_inset
and
\begin_inset Formula $k$
\end_inset
, the former case can happen only if
\begin_inset Formula $k=0$
\end_inset
and the latter only if
\begin_inset Formula $c=0\wedge k_{0}=k$
\end_inset
.
\begin_inset Formula
\begin{eqnarray*}
\left|\ph\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right|<\pi & \Leftrightarrow & \left|\ph\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right|\neq\pi\\
\varphi & \equiv & \ph\left(c-ik_{0}\right)<0,\\
\ph k & \equiv & 0\\
\ph\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}} & = & 2\varphi\\
\rightsquigarrow\left|\varphi\right| & \neq & \pi/2\\
\rightsquigarrow c & \neq & k_{0}\\
\left|\ph\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)\right| & = & \left|-2\varphi+\ph\left(\left(c-ik_{0}\right)^{2}+k^{2}\right)\right|
\end{eqnarray*}
\end_inset
Finally, swapping the first two arguments of
\begin_inset Formula $\hgfr$
\end_inset
in the hypergeometric represenation [REF DLMF 14.3.6] (note [REF DLMF §14.21(iii)]
that this also holds for complex arguments) of Legendre functions gives
\begin_inset Formula $P_{\nu}^{\mu}=P_{-\nu-1}^{\mu}$
\end_inset
, so the above result can be written
\begin_inset Formula
\[
\pht n{s_{q,k_{0}}^{\textup{L}1,c}}\left(k\right)=\frac{k^{n}\text{Γ}\left(2-q+n\right)}{k_{0}^{q}\left(c-ik_{0}\right)^{2-q+n}}\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{-\frac{n}{2}}\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{\frac{q}{2}-1}P_{q}^{-n}\left(\frac{1}{\sqrt{1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}}}\right).
\]
\end_inset
Let's polish it a bit more
\begin_inset Formula
\begin{eqnarray*}
\pht n{s_{q,k_{0}}^{\textup{L}1,c}}\left(k\right) & = & \frac{Γ\left(2-q+n\right)}{k_{0}^{q}\left(c-ik_{0}\right)^{2-q}}\left(-1\right)^{-\frac{n}{2}}\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{\frac{q}{2}-1}P_{q}^{-n}\left(\frac{1}{\sqrt{1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}}}\right)\\
& = & \frac{\text{Γ}\left(2-q+n\right)}{k_{0}^{q}}\left(-1\right)^{-\frac{n}{2}}\left(\left(c-ik_{0}\right)^{2}+k^{2}\right)^{\frac{q}{2}-1}P_{q}^{-n}\left(\frac{1}{\sqrt{1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}}}\right).
\end{eqnarray*}
\end_inset
\end_layout
\end_inset
\size footnotesize
\begin_inset Formula
\begin{multline}
\pht n{s_{q,k_{0}}^{\textup{L}1,c}}\left(k\right)=\frac{k^{n}Γ\left(2-q+n\right)}{k_{0}^{q}\left(c-ik_{0}\right)^{2-q+n}}\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{-\frac{n}{2}}\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{\frac{q}{2}-1}P_{q}^{-n}\left(\frac{1}{\sqrt{1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}}}\right),\\
k>0\wedge k_{0}>0\wedge c\ge0\wedge\lnot\left(c=0\wedge k_{0}=k\right)\label{eq:2D Hankel transform of exponentially suppressed outgoing wave expanded}
\end{multline}
\end_inset
\size default
with principal branches of the hypergeometric functions, associated Legendre
functions, and fractional powers.
The conditions from
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:2D Hankel transform of exponentially suppressed outgoing wave as 2F1"
\end_inset
should hold, but we will use
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:2D Hankel transform of exponentially suppressed outgoing wave expanded"
\end_inset
formally even if they are violated, with the hope that the divergences
eventually cancel in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:2D Hankel transform of regularized outgoing wave, decomposition"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Note Note
status collapsed
\begin_layout Plain Layout
Let's do it.
\begin_inset Formula
\begin{eqnarray*}
\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\left(\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{n}{2}}\left(1+\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{\frac{q}{2}-1}P_{q}^{-n}\left(\frac{1}{\sqrt{1+\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}}}\right)\\
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\left(\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{n}{2}}\left(1+\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{\frac{q}{2}-1}P_{q}^{-n}\left(\frac{1}{\sqrt{1+\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}}}\right)
\end{eqnarray*}
\end_inset
\end_layout
\end_inset
One problematic element here is the gamma function
\begin_inset Formula $\text{Γ}\left(2-q+n\right)$
\end_inset
which is singular if the arguments are negative integers, i.e.
if
\begin_inset Formula $q-n\ge3$
\end_inset
; but at least the necessary minimum of
\begin_inset Formula $q=1,2$
\end_inset
would be covered this way.
The associated Legendre function can be expressed as a finite
\begin_inset Quotes eld
\end_inset
polynomial
\begin_inset Quotes erd
\end_inset
if
\begin_inset Formula $q\ge n$
\end_inset
.
In other cases, different expressions can be obtained from
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:2D Hankel transform of exponentially suppressed outgoing wave as 2F1"
\end_inset
using various transformation formulae from either DLMF or
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{russian}
\end_layout
\end_inset
Прудников
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{russian}
\end_layout
\end_inset
.
\end_layout
\begin_layout Standard
In fact, Mathematica is usually able to calculate the transforms for specific
values of
\begin_inset Formula $\kappa,q,n$
\end_inset
, but it did not find any general formula for me.
The resulting expressions are finite sums of algebraic functions, Table
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Asymptotical-behaviour-Mathematica"
\end_inset
shows how fast they decay with growing
\begin_inset Formula $k$
\end_inset
for some parameters.
The only case where Mathematica did not help at all is
\begin_inset Formula $q=2,n=0$
\end_inset
, which is unfortunately important.
But if I have not made some mistake, the expression
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:2D Hankel transform of exponentially suppressed outgoing wave expanded"
\end_inset
is applicable for this case.
\end_layout
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\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
2
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
x
\end_layout
\end_inset
</cell>
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\begin_layout Plain Layout
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3
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2
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\size footnotesize
1
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\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Asymptotical behaviour of some
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:2D Hankel transform of regularized outgoing wave, decomposition"
\end_inset
obtained by Mathematica for
\begin_inset Formula $k\to\infty$
\end_inset
.
The table entries are the
\begin_inset Formula $N$
\end_inset
of
\begin_inset Formula $\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right)=o\left(1/k^{N}\right)$
\end_inset
.
The special entry
\begin_inset Quotes eld
\end_inset
x
\begin_inset Quotes erd
\end_inset
means that Mathematica was not able to calculate the integral, and
\begin_inset Quotes eld
\end_inset
w
\begin_inset Quotes erd
\end_inset
denotes that the first returned term was not simply of the kind
\begin_inset Formula $(\ldots)k^{-N-1}$
\end_inset
.
\begin_inset CommandInset label
LatexCommand label
name "tab:Asymptotical-behaviour-Mathematica"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
From Wikipedia page on binomial coefficient, eq.
(10) and around:
\end_layout
\begin_layout Plain Layout
When
\begin_inset Formula $P(x)$
\end_inset
is of degree less than or equal to
\begin_inset Formula $n$
\end_inset
,
\begin_inset Formula
\[
\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}P(n-j)=n!a_{n}
\]
\end_inset
where
\begin_inset Formula $a_{n}$
\end_inset
is the coefficient of degree
\begin_inset Formula $n$
\end_inset
in
\begin_inset Formula $P(x)$
\end_inset
.
\end_layout
\begin_layout Plain Layout
More generally,
\begin_inset Formula
\[
\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}P(m+(n-j)d)=d^{n}n!a_{n}
\]
\end_inset
where
\begin_inset Formula $m$
\end_inset
and
\begin_inset Formula $d$
\end_inset
are complex numbers.
\end_layout
\end_inset
\begin_inset Note Note
status open
\begin_layout Subsubsection
Hankel transforms of the long-range parts, alternative regularisation with
\begin_inset Formula $e^{-p/x^{2}}$
\end_inset
\begin_inset CommandInset label
LatexCommand label
name "sub:Hankel-transforms-ig-reg"
\end_inset
\end_layout
\begin_layout Plain Layout
From [REF
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{russian}
\end_layout
\end_inset
Прудников, том 2
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{russian}
\end_layout
\end_inset
, 2.12.9.14]
\begin_inset Formula
\begin{multline}
\int_{0}^{\infty}x^{\alpha-1}e^{-p/x^{2}}J_{\nu}\left(cx\right)\,\ud x=\frac{2^{\alpha-1}}{c^{\alpha}}Γ\begin{bmatrix}\left(\alpha+\nu\right)/2\\
1+\left(\nu-\alpha\right)/2
\end{bmatrix}{}_{0}F_{2}\left(1-\frac{\nu+\alpha}{2},1+\frac{\nu-\alpha}{2};\frac{c^{2}p}{4}\right)\\
+\frac{c^{\nu}p^{\left(\alpha+\nu\right)/2}}{2^{\nu+1}}\text{Γ}\begin{bmatrix}\left(\alpha+\nu\right)/2\\
\nu+1
\end{bmatrix}{}_{0}F_{2}\left(1+\frac{\nu+\alpha}{2},\nu+1;\frac{c^{2}p}{4}\right),\qquad[c,\Re p>0;\Re\alpha<3/2].\label{eq:prudnikov2 eq 2.12.9.14}
\end{multline}
\end_inset
Let now
\begin_inset Formula $\rho_{p}^{\textup{ig.}}$
\end_inset
from
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:inverse gaussian regularisation function"
\end_inset
serve as the regularisation fuction and
\begin_inset Formula
\[
\pht n{s_{q,k_{0}}^{\textup{L}'p}}\left(k\right)\equiv\int_{0}^{\infty}\frac{e^{ik_{0}r}}{\left(k_{0}r\right)^{q}}J_{n}\left(kr\right)e^{-p/r^{2}}r\,\ud r.
\]
\end_inset
And it seems that this is a dead-end, because
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:prudnikov2 eq 2.12.9.14"
\end_inset
cannot deal with the
\begin_inset Formula $e^{ik_{0}r}$
\end_inset
part.
Damn.
\end_layout
\end_inset
\end_layout
\begin_layout Subsection
3d (TODO)
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{multline*}
\uaft{S_{l',m',t'\leftarrow l,m,t}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\\
\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect k},\phi_{\vect k}\right)\left(-i\right)^{p}\usht p{z_{p}^{(J)}}\left(\left|\vect k\right|\right)
\end{multline*}
\end_inset
\end_layout
\begin_layout Section
Major TODOs and open questions
\end_layout
\begin_layout Itemize
Check if
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:2D Hankel transform of exponentially suppressed outgoing wave expanded"
\end_inset
gives a satisfactory result for the case
\begin_inset Formula $q=2,n=0$
\end_inset
.
\end_layout
\begin_layout Itemize
Analyse the behaviour
\begin_inset Formula $k\to k_{0}$
\end_inset
.
\end_layout
\begin_layout Itemize
Find a general algorithm for generating the expressions of the Hankel transforms.
\end_layout
\begin_layout Itemize
Three-dimensional case.
\end_layout
\begin_layout Section
(Appendix) Fourier vs.
Hankel transform
\end_layout
\begin_layout Subsection
Three dimensions
\end_layout
\begin_layout Standard
Given a nice enough function
\begin_inset Formula $f$
\end_inset
of a real 3d variable, assume its factorisation into radial and angular
parts
\begin_inset Formula
\[
f(\vect r)=\sum_{l,m}f_{l,m}(\left|\vect r\right|)\ush lm\left(\theta_{\vect r},\phi_{\vect r}\right).
\]
\end_inset
Acording to (REF Baddour 2010, eqs.
13, 16), its Fourier transform can then be expressed in terms of Hankel
transforms (CHECK normalisation of
\begin_inset Formula $j_{n}$
\end_inset
, REF Baddour (1))
\begin_inset Formula
\[
\uaft f(\vect k)=\frac{4\pi}{\left(2\pi\right)^{\frac{3}{2}}}\sum_{l,m}\left(-i\right)^{l}\left(\bsht{f_{l,m}}{}\right)\left(\left|\vect k\right|\right)\ush lm\left(\theta_{\vect k},\phi_{\vect k}\right)
\]
\end_inset
where the spherical Hankel transform
\begin_inset Formula $\bsht l{}$
\end_inset
of degree
\begin_inset Formula $l$
\end_inset
is defined as (REF Baddour eq.
2)
\begin_inset Formula
\[
\bsht lg(k)\equiv\int_{0}^{\infty}\ud r\, r^{2}g(r)j_{l}\left(kr\right).
\]
\end_inset
Using this convention, the inverse spherical Hankel transform is given by
(REF Baddour eq.
3)
\begin_inset Formula
\[
g(r)=\frac{2}{\pi}\int_{0}^{\infty}\ud k\, k^{2}\bsht lg(k)j_{l}(k),
\]
\end_inset
so it is not unitary.
\end_layout
\begin_layout Standard
An unitary convention would look like this:
\begin_inset Formula
\begin{equation}
\usht lg(k)\equiv\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\ud r\, r^{2}g(r)j_{l}\left(kr\right).\label{eq:unitary 3d Hankel tf definition}
\end{equation}
\end_inset
Then
\begin_inset Formula $\usht l{}^{-1}=\usht l{}$
\end_inset
and the unitary, angular-momentum Fourier transform reads
\begin_inset Formula
\begin{eqnarray}
\uaft f(\vect k) & = & \frac{4\pi}{\left(2\pi\right)^{\frac{3}{2}}}\sqrt{\frac{\pi}{2}}\sum_{l,m}\left(-i\right)^{l}\left(\usht l{f_{l,m}}\right)\left(\left|\vect k\right|\right)\ush lm\left(\theta_{\vect k},\phi_{\vect k}\right)\nonumber \\
& = & \sum_{l,m}\left(-i\right)^{l}\left(\usht l{f_{l,m}}\right)\left(\left|\vect k\right|\right)\ush lm\left(\theta_{\vect k},\phi_{\vect k}\right).\label{eq:Fourier v. Hankel tf 3d}
\end{eqnarray}
\end_inset
Cool.
\end_layout
\begin_layout Subsection
Two dimensions
\end_layout
\begin_layout Standard
Similarly in 2d, let the expansion of
\begin_inset Formula $f$
\end_inset
be
\begin_inset Formula
\[
f\left(\vect r\right)=\sum_{m}f_{m}\left(\left|\vect r\right|\right)e^{im\phi_{\vect r}},
\]
\end_inset
its Fourier transform is then (CHECK this, it is taken from the Wikipedia
article on Hankel transform)
\begin_inset Formula
\begin{equation}
\uaft f\left(\vect k\right)=\sum_{m}i^{m}e^{im\phi_{\vect k}}\pht mf_{m}\left(\left|\vect k\right|\right)\label{eq:Fourier v. Hankel tf 2d}
\end{equation}
\end_inset
where the Hankel transform of order
\begin_inset Formula $m$
\end_inset
is defined as
\begin_inset Formula
\begin{equation}
\pht mg\left(k\right)=\int_{0}^{\infty}\ud r\, g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition}
\end{equation}
\end_inset
which is already self-inverse,
\begin_inset Formula $\pht m{}^{-1}=\pht m{}$
\end_inset
(hence also unitary).
\end_layout
\begin_layout Section
(Appendix) Multidimensional Dirac comb
\end_layout
\begin_layout Subsection
1D
\end_layout
\begin_layout Standard
This is all from Wikipedia
\end_layout
\begin_layout Subsubsection
Definitions
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{eqnarray*}
Ш(t) & \equiv & \sum_{k=-\infty}^{\infty}\delta(t-k)\\
Ш_{T}(t) & \equiv & \sum_{k=-\infty}^{\infty}\delta(t-kT)=\frac{1}{T}Ш\left(\frac{t}{T}\right)
\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Subsubsection
Fourier series representation
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
Ш_{T}(t)=\sum_{n=-\infty}^{\infty}e^{2\pi int/T}\label{eq:1D Dirac comb Fourier series}
\end{equation}
\end_inset
\end_layout
\begin_layout Subsubsection
Fourier transform
\end_layout
\begin_layout Standard
With unitary ordinary frequency Ft., i.e.
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\uoft f(\vect{\xi})\equiv\int_{\mathbb{R}^{n}}f(\vect x)e^{-2\pi i\vect x\cdot\vect{\xi}}\ud^{n}\vect x
\]
\end_inset
we have
\begin_inset Formula
\begin{equation}
\uoft{Ш_{T}}(f)=\frac{1}{T}Ш_{\frac{1}{T}}(f)=\sum_{n=-\infty}^{\infty}e^{-i2\pi fnT}\label{eq:1D Dirac comb Ft ordinary freq}
\end{equation}
\end_inset
and with unitary angular frequency Ft., i.e.
\begin_inset Formula
\begin{equation}
\uaft f(\vect k)\equiv\frac{1}{\left(2\pi\right)^{n/2}}\int_{\mathbb{R}^{n}}f(\vect x)e^{-i\vect x\cdot\vect k}\ud^{n}\vect x\label{eq:Ft unitary angular frequency}
\end{equation}
\end_inset
we have (CHECK)
\begin_inset Formula
\[
\uaft{Ш_{T}}(\omega)=\frac{\sqrt{2\pi}}{T}Ш_{\frac{2\pi}{T}}(\omega)=\frac{1}{\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}e^{-i\omega nT}
\]
\end_inset
\end_layout
\begin_layout Subsection
Dirac comb for multidimensional lattices
\end_layout
\begin_layout Subsubsection
Definitions
\end_layout
\begin_layout Standard
Let
\begin_inset Formula $d$
\end_inset
be the dimensionality of the real vector space in question, and let
\begin_inset Formula $\basis u\equiv\left\{ \vect u_{i}\right\} _{i=1}^{d}$
\end_inset
denote a basis for some lattice in that space.
Let the corresponding lattice delta comb be
\begin_inset Formula
\[
\dc{\basis u}\left(\vect x\right)\equiv\sum_{n_{1}=-\infty}^{\infty}\ldots\sum_{n_{d}=-\infty}^{\infty}\delta\left(\vect x-\sum_{i=1}^{d}n_{i}\vect u_{i}\right).
\]
\end_inset
\end_layout
\begin_layout Standard
Furthemore, let
\begin_inset Formula $\rec{\basis u}\equiv\left\{ \rec{\vect u}_{i}\right\} _{i=1}^{d}$
\end_inset
be the reciprocal lattice basis, that is the basis satisfying
\begin_inset Formula $\vect u_{i}\cdot\rec{\vect u_{j}}=\delta_{ij}$
\end_inset
.
This slightly differs from the usual definition of a reciprocal basis,
here denoted
\begin_inset Formula $\recb{\basis u}\equiv\left\{ \recb{\vect u_{i}}\right\} _{i=1}^{d}$
\end_inset
, which satisfies
\begin_inset Formula $\vect u_{i}\cdot\recb{\vect u_{j}}=2\pi\delta_{ij}$
\end_inset
instead.
\end_layout
\begin_layout Subsubsection
Factorisation of a multidimensional lattice delta comb
\end_layout
\begin_layout Standard
By simple drawing, it can be seen that
\begin_inset Formula
\[
\dc{\basis u}(\vect x)=c_{\basis u}\prod_{i=1}^{d}\dc{}\left(\vect x\cdot\rec{\vect u_{i}}\right)
\]
\end_inset
where
\begin_inset Formula $c_{\basis u}$
\end_inset
is some numerical volume factor.
In order to determine
\begin_inset Formula $c_{\basis u}$
\end_inset
, let us consider only the
\begin_inset Quotes eld
\end_inset
zero tooth
\begin_inset Quotes erd
\end_inset
of the comb, leading to
\begin_inset Formula
\[
\delta^{d}(\vect x)=c_{\basis u}\prod_{i=1}^{d}\delta\left(\vect x\cdot\rec{\vect u_{i}}\right).
\]
\end_inset
From the scaling property of delta function,
\begin_inset Formula $\delta(ax)=\left|a\right|^{-1}\delta(x)$
\end_inset
, we get
\begin_inset Formula
\[
\delta^{d}(\vect x)=c_{\basis u}\prod_{i=1}^{d}\left\Vert \rec{\vect u_{i}}\right\Vert ^{-1}\delta\left(\vect x\cdot\frac{\rec{\vect u_{i}}}{\left\Vert \rec{\vect u_{i}}\right\Vert }\right).
\]
\end_inset
\end_layout
\begin_layout Standard
From the Osgood's book (p.
375):
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\dc A(\vect x)=\frac{1}{\left|\det A\right|}\dc{}^{(d)}\left(A^{-1}\vect x\right)
\]
\end_inset
\begin_inset Note Note
status open
\begin_layout Plain Layout
Applying both sides to a test function that is one at the origin, we get
\begin_inset Formula $c_{\basis u}=\prod_{i=1}^{d}\left\Vert \rec{\vect u_{i}}\right\Vert $
\end_inset
SRSLY?, and hence
\begin_inset Formula
\begin{equation}
\dc{\basis u}(\vect x)=\prod_{i=1}^{d}\left\Vert \rec{\vect u_{i}}\right\Vert \dc{}\left(\vect x\cdot\rec{\vect u_{i}}\right).\label{eq:Dirac comb factorisation}
\end{equation}
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Subsubsection
Fourier series representation
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
Utilising the Fourier series for 1D Dirac comb
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:1D Dirac comb Fourier series"
\end_inset
and the factorisation
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Dirac comb factorisation"
\end_inset
, we get
\begin_inset Formula
\begin{eqnarray*}
\dc{\basis u}(\vect x) & = & \prod_{j=1}^{d}\left\Vert \rec{\vect u_{j}}\right\Vert \sum_{n_{j}=-\infty}^{\infty}e^{2\pi in_{i}\vect x\cdot\rec{\vect u_{i}}}\\
& = & \left(\prod_{j=1}^{d}\left\Vert \rec{\vect u_{j}}\right\Vert \right)\sum_{\vect n\in\mathbb{Z}^{d}}e^{2\pi i\vect x\cdot\sum_{k=1}^{d}n_{k}\rec{\vect u_{k}}}.
\end{eqnarray*}
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Subsubsection
Fourier transform (OK)
\end_layout
\begin_layout Standard
From the Osgood's book https://see.stanford.edu/materials/lsoftaee261/chap8.pdf,
p.
379
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\uoft{\dc{\basis u}}\left(\vect{\xi}\right)=\left|\det\rec{\basis u}\right|\dc{\rec{\basis u}}^{(d)}\left(\vect{\xi}\right).
\]
\end_inset
And consequently, for unitary/angular frequency it is
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{eqnarray}
\uaft{\dc{\basis u}}\left(\vect k\right) & = & \frac{1}{\left(2\pi\right)^{\frac{d}{2}}}\uoft{\dc{\basis u}}\left(\frac{\vect k}{2\pi}\right)\nonumber \\
& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\dc{\rec{\basis u}}^{(d)}\left(\frac{\vect k}{2\pi}\right)\nonumber \\
& = & \left(2\pi\right)^{\frac{d}{2}}\left|\det\rec{\basis u}\right|\dc{\recb{\basis u}}\left(\vect k\right)\nonumber \\
& = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\dc{\recb{\basis u}}\left(\vect k\right).\label{eq:Dirac comb uaFt}
\end{eqnarray}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
On the third line, we used the stretch theorem, getting
\begin_inset Formula
\[
\dc{\recb{\basis u}}\left(\vect k\right)=\dc{2\pi\rec{\basis u}}\left(\vect k\right)=\left(2\pi\right)^{-d}\dc{\rec{\basis u}}\left(\frac{\vect k}{2\pi}\right)
\]
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Subsubsection
Convolution
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\left(f\ast\dc{\basis u}\right)(\vect x)=\sum_{\vect t\in\basis u\ints^{d}}f(\vect x-\vect t)
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
So, from the stretch theorem
\begin_inset Formula $\uoft{(f(A\vect x))}=\frac{1}{\left|\det A\right|}\uoft{f\left(A^{-T}\vect{\xi}\right)}=\left|\det A^{-T}\right|\uoft{f\left(A^{-T}\vect{\xi}\right)}$
\end_inset
\end_layout
\begin_layout Plain Layout
From
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Dirac comb factorisation"
\end_inset
and
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:1D Dirac comb Ft ordinary freq"
\end_inset
\begin_inset Formula
\[
\uoft{\dc{\basis u}}(\vect{\xi})=\prod_{i=1}^{d}\left\Vert \rec{\vect u_{i}}\right\Vert \dc{}\left(\vect x\cdot\rec{\vect u_{i}}\right).
\]
\end_inset
\end_layout
\end_inset
\end_layout
\end_body
\end_document